Re: Incomplete Gamma function
- To: mathgroup at smc.vnet.net
- Subject: [mg67654] Re: Incomplete Gamma function
- From: "James Gilmore" <james.gilmore at yale.edu>
- Date: Mon, 3 Jul 2006 06:38:39 -0400 (EDT)
- Organization: Yale University
- References: <F1CC734F-5AD5-446A-ACF9-14487B8D6637@mimuw.edu.pl> <e888ar$930$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hi Andrzej, This issue has been reported previously... in 2003 when I came across it working on high order hydrogenic wave/greens functions with Mathematica 4.2 and 5.0. Lets hope for a amendment in 6?? In[234]:= FunctionExpand[Gamma[14, z]] FunctionExpand[Gamma[15, z]] $Version Out[234]= z^13/E^z + 13*(z^12/E^z + 12*(z^11/E^z + 11*(z^10/E^z + 10*(z^9/E^z + 9*(z^8/E^z + 8*(z^7/E^z + 7*(z^6/E^z + 6*(z^5/E^z + 5*(z^4/E^z + 4*(z^3/E^z + 3*(z^2/E^z + 2*(E^(-z) + z/E^z)))))))))))) Out[235]= Gamma[15, z] Out[236]= "5.2 for Microsoft Windows (June 20, 2005)" Cheers James Gilmore "Andrzej Kozlowski" <akoz at mimuw.edu.pl> wrote in message news:e888ar$930$1 at smc.vnet.net... >I have found the answer to the question myself, less than 5 minutes > after posting it... > Looking at > > http://mathworld.wolfram.com/IncompleteGammaFunction.html > > I see that, for an integer n we have: > > Gamma[n-1,n] = (n-1)! Sum[(n-1)^k/k!,{k,0,n-1}] > > It is completely obvious that this is always an integer. So now the > question is: why doesn't Mathematica (or FunctionExpand) make use of > this formula to compute Gamma[n,x] for integer n? > > Andrzej Kozlowski > > > > On 2 Jul 2006, at 11:16, Andrzej Kozlowski wrote: > >> Here is a question I have come across concerning special >> functions, which unfortunately, is an area of mathematics I know >> next to nothing about. However, as there are many experts in this >> field who read this list I hope someone will find this an >> interesting (or trivial?) question. >> >> Consider the following sequence: >> >> >> Table[FunctionExpand[Exp[n-1]*Gamma[n,n-1]],{n,2,14}] >> >> {2,10,78,824,10970,176112,3309110,71219584,1727242866,46602156800, >> 1384438376222,44902138752000,1578690429731402} >> >> As you see, we get only integers. What happens if n is larger than >> 14? Mathematica seems not to be able to answer this: >> >> >> FunctionExpand[Exp[n - 1]* >> Gamma[n, n - 1]] /. >> n -> 15 >> >> >> E^14*Gamma[15, 14] >> >> Numerical methods also do not seem to be able to determine if this >> is an integer or not. I have looked at Abramowitz & Stegun but I >> can't see anything that obviously helps to resolve the issue. Can >> anyone help? >> >> Andrzej Kozlowski >> >> Tokyo, Japan >